Non-Euclidean Statistics Building on Generalized Fréchet Means and Applications

Abstract

Fréchet extended the concept of the expected value of a random variable
in a Euclidean space to the minimizer of expected squared distance on a
metric space. If the metric space is a manifold, the concept of
covariance or principal compontents generalizes similarly, but slightly
less canonically to best approximating geodesics, or best approximating
subspaces subject to some restrictions. We introduce such generalized
Fréchet means on stratified spaces, investigate some of their surprising
asymptotic properties and give applications to recent research on
micromolecular structure and phylogenetics.